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arXiv:2202.04314 · Physical Review Letters

Entanglement of Nambu Spinors and Bell Inequality Test without Beam Splitters

Wei Luo, Hao Geng, D. Y. Xing, G. Blatter, Wei Chen

Published Sep 16, 2022• 7 citations
arXivHTMLPDF

Abstract

The identification of electronic entanglement in solids remains elusive so far, which is owed to the difficulty of implementing spinor-selective beam splitters with tunable polarization direction. Here, we propose to overcome this obstacle by producing and detecting a particular type of entanglement encoded in the Nambu spinor or electron-hole components of quasiparticles excited in quantum Hall edge states. Because of the opposite charge of electrons and holes, the detection of the Nambu spinor translates into a charge-current measurement, which eliminates the need for beam splitters and assures a high detection rate. Conveniently, the spinor correlation function at fixed effective polarizations derives from a single current-noise measurement, with the polarization directions of the detector easily adjusted by coupling the edge states to a voltage gate and a superconductor, both having been realized in experiments. We show that the violation of Bell inequality occurs in a large parameter region. Our Letter opens a new route for probing quasiparticle entanglement in solid-state physics exempt from traditional beam splitters.

DOI
10.1103/physrevlett.129.120507

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The identification of electronic entanglement in solids remains elusive so far, which is owed to the difficulty of implementing spinor-selective beam splitters with tunable polarization direction. Here, we propose to overcome this obstacle by producing and detecting a particular type of entanglement encoded in the Nambu spinor or electron-hole components of quasiparticles excited in quantum Hall edge states. Because of the opposite charge of electrons and holes, the detection of the Nambu spinor translates into a charge-current measurement, which eliminates the need for beam splitters and assures a high detection rate. Conveniently, the spinor correlation function at fixed effective polarizations derives from a single current-noise measurement, with the polarization directions of the detector easily adjusted by coupling the edge states to a voltage gate and a superconductor, both having been realized in experiments. We show that the violation of Bell inequality occurs in a large parameter region. Our Letter opens a new route for probing quasiparticle entanglement in solid-state physics exempt from traditional beam splitters.
Entanglement of spin degrees of freedom can drastically alter the orbital exchange symmetry of electrons, switching their bunching and antibunching behaviors and the resultant current correlations in the Hanbury-Brown-Twiss interferometry. Here, we investigate the exchange correlation of quasiparticles with entanglement encoded in the Nambu spinors, or the electron-hole degree of freedom. In contrast to the conventional correspondence between spin entanglement and current correlation, we find that singlet (triplet) entanglement of Nambu spinors results in suppressed (enhanced) current correlation. This effect arises because the charge degree of freedom itself encodes the entanglement. We propose implementing this phenomenon in the edge states of a quantum Hall system, where the entangled states of the Nambu spinors can be continuously tuned by gate voltages. Our study reveals an unconventional relationship between entanglement and charge correlations, offering an effective approach for detecting entanglement of Nambu spinors.
A Cooper-pair or electron-hole splitter is a device capable of spatially separating entangled fermionic quasiparticles into mesoscopic solid-state systems such as quantum dots or quantum wires. We theoretically study such a splitter based on a pair of helical Luttinger liquids, which arise naturally at the edges of a quantum spin Hall insulator. Equipping each helical liquid with a beam splitter, current-current cross correlations can be used to construct a Bell inequality whose violation would indicate nonlocal orbital entanglement of the injected electrons and/or holes. Due to the Luttinger-liquid correlations, however, the entanglement is suppressed depending on ambient temperature and voltage bias.
The generation of spin-entangled electrons is an important prerequisite for future solid-state quantum technologies. Cooper pairs in a superconductor can be split into separate electrons in a spin-singlet state, however, detecting their entanglement remains an open experimental challenge. Proposals to detect the entanglement by violating a Bell inequality typically require a large number of current cross-correlation measurements, and not all entangled states can be detected in this way. Here, we instead formulate an entanglement witness that can detect the spin-entanglement using only three cross-correlation measurements of the currents in the outputs of a Cooper pair splitter. We identify the optimal measurement settings for witnessing the entanglement, and we illustrate the use of our entanglement witness with a realistic model of a Cooper pair splitter for which we evaluate the cross-correlations of the output currents. Specifically, we find that the entanglement of the spins can be detected even with a moderate level of decoherence. Our work thereby paves the way for an experimental detection of the entanglement produced by Cooper pair splitters.
Abstract Correlations are fundamental in describing many body systems - not only in natural sciences. However, in experiments, correlations are notoriously difficult to assess on the microscopic scale, especially for electron spins. Here, we demonstrate a direct measurement of the spin cross-correlations between the currents of a Cooper pair splitter, an electronic device that emits electrons originating from Cooper pairs in a superconductor. While it is firmly established theoretically that these electron pairs form maximally spin-entangled singlet states with opposite spin projections, no spin correlation experiments have been demonstrated so far. We use ferromagnetic sidegates, compatible with superconducting electronic structures, to individually spin polarize the transmissions of two quantum dots fabricated in the two electronic paths, which act as tunable spin filters. The signals are detected in standard transport and in highly sensitive transconductance experiments. We find that the spin-cross correlation is negative, compatible with spin singlet emission, and deviates from the ideal value mostly due to a finite overlap of the Zeeman split quantum dot states. Our results demonstrate a new route to perform spin auto- and cross correlation experiments in nanometer scaled electronic devices, especially suitable for those relying on magnetic field sensitive superconducting elements, like unconventional, triplet or topologically non-trivial superconductors, or to perform Bell tests with massive particles, like electrons.
Spin-orbit interaction provides a spin filtering effect in carbon nanotube based Cooper pair splitters that allows us to determine spin correlators directly from current measurements. The spin filtering axes are tunable by a global external magnetic field. By a bending of the nanotube, the filtering axes on both sides of the Cooper pair splitter become sufficiently different that a test of entanglement of the injected Cooper pairs through a Bell-like inequality can be implemented. This implementation does not require noise measurements, supports imperfect splitting efficiency and disorder, and does not demand a full knowledge of the spin-orbit strength. Using a microscopic calculation we demonstrate that entanglement detection by violation of the Bell-like inequality is within the reach of current experimental setups.
Electron entanglement: splitting pairs One of the most counterintuitive aspects of quantum mechanics is non-locality, manifesting as spatially separated objects influencing each other in a direct way. Experimental verifications of such concepts have been conducted successfully using entangled pairs of photons to test so-called Bell inequalities, but similar demonstrations using electrons have remained elusive owing to the inherent difficulty of creating and splitting pairs of entangled electrons. Lukas Hofstetter and colleagues now demonstrate that this is possible in a Y-shaped electron entangler consisting of a superconductor at its stem, coupled via tunnelling barriers to two separate quantum dots along the fork-like branches of the device. The entangled pairs of electrons are naturally formed in the superconductor, which in its ground state consists of Cooper pairs of electrons, and split by Coulomb interactions and careful tuning of the energy levels of the quantum dots. The study paves the way for the study of electronic entanglement and experimental tests of Bell inequalities in the solid state.